This document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and total biomass as the focal ecosystem function.

Important terms:

  • Stage: With seed inflow, without seed inflow
  • Ninitial: Planted species richness

1 Grass1

Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.

1.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow                2.35      3.54    -4.37     9.40 1.00     2075     2352
## StageWithoutseedinflow           -12.45      4.02   -20.14    -4.11 1.00     2060     2210
## StageWithseedinflow:Shannon       17.76      1.49    14.81    20.62 1.00     2054     2012
## StageWithoutseedinflow:Shannon    25.79      1.98    21.77    29.54 1.00     2092     2050
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    23.18      0.61    22.00    24.37 1.00     2778     2345
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2971205 0.02313489 0.2487609 0.3401389

1.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

1.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

1.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

1.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


1.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                82.86     26.87    30.08   135.37 1.00     5734     2756
## Ninitial4:StageWithseedinflow               151.47     24.19   103.90   198.20 1.00     5389     2863
## Ninitial8:StageWithseedinflow               167.07     20.20   127.09   205.43 1.00     6224     2593
## Ninitial16:StageWithseedinflow              250.92     17.44   217.08   285.36 1.00     5264     2884
## Ninitial32:StageWithseedinflow              278.74     19.99   238.57   317.27 1.00     5573     3139
## Ninitial2:StageWithoutseedinflow             22.84     10.73     1.93    44.10 1.00     5833     2782
## Ninitial4:StageWithoutseedinflow             64.19     14.89    34.84    92.98 1.00     5809     3160
## Ninitial8:StageWithoutseedinflow             89.41     14.54    60.77   117.58 1.00     6116     2713
## Ninitial16:StageWithoutseedinflow           184.97     17.40   151.48   218.40 1.00     5330     2982
## Ninitial32:StageWithoutseedinflow           188.96     23.27   141.86   235.28 1.00     5239     2681
## Ninitial2:StageWithseedinflow:Shannon       -36.57     16.75   -69.24    -3.90 1.00     5746     2714
## Ninitial4:StageWithseedinflow:Shannon       -54.24     11.07   -75.38   -32.49 1.00     5402     2823
## Ninitial8:StageWithseedinflow:Shannon       -47.61      7.62   -62.29   -32.42 1.00     6253     2727
## Ninitial16:StageWithseedinflow:Shannon      -65.03      5.92   -76.67   -53.52 1.00     5202     3000
## Ninitial32:StageWithseedinflow:Shannon      -64.82      6.32   -76.99   -52.27 1.00     5541     3146
## Ninitial2:StageWithoutseedinflow:Shannon     -3.25      7.21   -17.53    10.74 1.00     5541     2709
## Ninitial4:StageWithoutseedinflow:Shannon    -19.14      7.64   -34.06    -4.10 1.00     5760     3188
## Ninitial8:StageWithoutseedinflow:Shannon    -24.07      6.44   -36.96   -11.38 1.00     6132     2732
## Ninitial16:StageWithoutseedinflow:Shannon   -52.87      7.18   -66.68   -38.79 1.00     5341     2858
## Ninitial32:StageWithoutseedinflow:Shannon   -44.10      8.87   -61.66   -26.34 1.00     5269     2522
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    15.05      0.43    14.22    15.93 1.00     6762     2838
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.6997435 0.01090152 0.6772719 0.7192029

1.2.1 Posterior predictive checks

1.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

1.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

1.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


2 Grass2

Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.

2.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               44.58      1.57    41.59    47.66 1.00     2205     2333
## StageWithoutseedinflow            37.48      1.77    34.09    40.95 1.00     1971     2216
## StageWithseedinflow:Shannon        8.69      0.56     7.60     9.78 1.00     2172     2362
## StageWithoutseedinflow:Shannon    15.09      0.78    13.54    16.61 1.00     1968     2086
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    12.22      0.32    11.61    12.86 1.00     2922     1948
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4411902 0.01957698 0.4012542 0.4774241

2.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

2.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

2.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

2.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


2.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                84.80     33.70    15.99   150.50 1.00     4189     2834
## Ninitial4:StageWithseedinflow               136.13     36.43    63.04   209.12 1.00     4721     2397
## Ninitial8:StageWithseedinflow               133.23     31.32    72.85   195.25 1.00     4742     2594
## Ninitial16:StageWithseedinflow               91.96     50.76    -8.14   191.89 1.00     4183     2514
## Ninitial32:StageWithseedinflow               42.94     62.67   -80.61   166.86 1.00     4230     2509
## Ninitial2:StageWithoutseedinflow            -16.58     22.66   -61.55    27.91 1.00     4407     2901
## Ninitial4:StageWithoutseedinflow             27.23     13.63     0.17    54.14 1.00     4674     2684
## Ninitial8:StageWithoutseedinflow             38.60     10.78    16.71    59.77 1.00     4472     2791
## Ninitial16:StageWithoutseedinflow            47.47     11.41    25.41    70.15 1.00     4346     2405
## Ninitial32:StageWithoutseedinflow            72.56     14.83    42.74   101.49 1.00     5203     2906
## Ninitial2:StageWithseedinflow:Shannon       -16.03     20.42   -56.08    25.88 1.00     4196     2885
## Ninitial4:StageWithseedinflow:Shannon       -29.84     15.80   -61.57     1.86 1.00     4693     2283
## Ninitial8:StageWithseedinflow:Shannon       -20.14     10.81   -41.53     0.55 1.00     4729     2636
## Ninitial16:StageWithseedinflow:Shannon       -4.55     14.33   -32.79    23.69 1.00     4188     2600
## Ninitial32:StageWithseedinflow:Shannon        8.12     14.93   -21.45    37.48 1.00     4227     2536
## Ninitial2:StageWithoutseedinflow:Shannon     46.13     14.00    18.44    73.91 1.00     4407     2985
## Ninitial4:StageWithoutseedinflow:Shannon     22.09      7.40     7.47    36.81 1.00     4722     2772
## Ninitial8:StageWithoutseedinflow:Shannon     17.72      4.88     8.00    27.67 1.00     4500     2836
## Ninitial16:StageWithoutseedinflow:Shannon    12.22      4.17     3.96    20.41 1.00     4316     2456
## Ninitial32:StageWithoutseedinflow:Shannon     3.26      4.42    -5.40    12.06 1.00     5192     2863
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.04      0.26     8.56     9.58 1.00     4826     2528
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate Est.Error      Q2.5     Q97.5
## R2 0.4941592 0.0196986 0.4523918 0.5304386

2.2.1 Posterior predictive checks

2.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

2.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

2.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


3 Grass3

May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.

3.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               46.11      1.45    43.30    49.06 1.00     1873     1957
## StageWithoutseedinflow            46.00      1.44    43.27    48.83 1.00     1840     2019
## StageWithseedinflow:Shannon        2.68      0.54     1.62     3.72 1.00     1909     1824
## StageWithoutseedinflow:Shannon     2.88      0.57     1.75     3.99 1.00     1809     1885
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.96      0.25     9.48    10.47 1.00     2844     2314
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5     Q97.5
## R2 0.06589133 0.01661286 0.03520367 0.1002929

3.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


3.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                18.41     36.96   -51.87    91.92 1.00     4129     3049
## Ninitial4:StageWithseedinflow                -7.05     25.73   -58.38    43.21 1.00     3615     2618
## Ninitial8:StageWithseedinflow                 4.23     26.82   -48.98    55.79 1.00     3929     2779
## Ninitial16:StageWithseedinflow               58.85     22.61    14.16   102.76 1.00     3958     2928
## Ninitial32:StageWithseedinflow               48.93     30.93    -9.71   110.08 1.00     3786     2671
## Ninitial2:StageWithoutseedinflow             64.95      6.46    51.66    77.13 1.00     3404     2608
## Ninitial4:StageWithoutseedinflow             25.40     11.37     3.01    47.62 1.00     3724     2853
## Ninitial8:StageWithoutseedinflow             37.27     11.59    14.76    60.20 1.00     3988     2779
## Ninitial16:StageWithoutseedinflow            65.16     21.11    24.61   105.65 1.00     3614     3027
## Ninitial32:StageWithoutseedinflow            78.89     32.49    15.88   144.36 1.00     3958     2597
## Ninitial2:StageWithseedinflow:Shannon        18.29     22.15   -26.03    60.63 1.00     4138     3035
## Ninitial4:StageWithseedinflow:Shannon        25.10     11.26     3.31    47.44 1.00     3609     2712
## Ninitial8:StageWithseedinflow:Shannon        17.30      9.48    -0.93    36.05 1.00     3925     2845
## Ninitial16:StageWithseedinflow:Shannon       -1.06      6.70   -14.00    12.32 1.00     3971     2962
## Ninitial32:StageWithseedinflow:Shannon        2.32      8.12   -13.70    17.62 1.00     3793     2670
## Ninitial2:StageWithoutseedinflow:Shannon    -10.02      4.01   -17.62    -1.76 1.00     3564     2543
## Ninitial4:StageWithoutseedinflow:Shannon     11.64      5.30     1.14    22.04 1.00     3733     2754
## Ninitial8:StageWithoutseedinflow:Shannon      6.02      4.42    -2.80    14.59 1.00     3986     2877
## Ninitial16:StageWithoutseedinflow:Shannon    -3.50      6.64   -16.25     9.30 1.00     3586     2947
## Ninitial32:StageWithoutseedinflow:Shannon    -5.77      9.23   -24.34    12.08 1.00     3962     2709
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     6.73      0.19     6.38     7.11 1.00     6008     3209
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2335496 0.02541694 0.1833591 0.2822491

3.2.1 Posterior predictive checks

3.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

3.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


4 Forest1

Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.

4.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               28.10      3.29    21.42    34.39 1.00     2109     1828
## StageWithoutseedinflow            30.66      3.78    23.41    38.21 1.00     2115     1956
## StageWithseedinflow:Shannon       15.01      1.97    11.22    19.03 1.00     2159     1990
## StageWithoutseedinflow:Shannon    13.86      2.78     8.43    19.19 1.00     2176     1945
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    21.89      0.55    20.81    23.01 1.00     2831     2463
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error       Q2.5     Q97.5
## R2 0.1043157 0.01906257 0.06958144 0.1435041

4.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


4.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                67.27     11.62    44.76    89.81 1.00     4244     2814
## Ninitial4:StageWithseedinflow                89.48      9.92    69.83   109.16 1.00     4379     2990
## Ninitial8:StageWithseedinflow                94.54     11.13    71.92   116.15 1.00     4154     2933
## Ninitial16:StageWithseedinflow               78.19     14.14    50.21   106.21 1.00     4285     2949
## Ninitial32:StageWithseedinflow               41.75     20.87     0.50    82.51 1.00     4315     2730
## Ninitial2:StageWithoutseedinflow             65.98     14.28    37.44    93.33 1.00     3789     2927
## Ninitial4:StageWithoutseedinflow             75.19      9.14    57.23    93.65 1.00     4670     3151
## Ninitial8:StageWithoutseedinflow            100.81     10.09    81.03   119.87 1.00     3594     2916
## Ninitial16:StageWithoutseedinflow            58.17      9.58    39.00    76.69 1.00     4101     2882
## Ninitial32:StageWithoutseedinflow            46.75      8.56    29.90    63.39 1.00     4599     3220
## Ninitial2:StageWithseedinflow:Shannon       -24.74      9.70   -43.61    -5.58 1.00     4101     2877
## Ninitial4:StageWithseedinflow:Shannon       -28.90      7.40   -43.83   -14.26 1.00     4337     2945
## Ninitial8:StageWithseedinflow:Shannon       -23.11      7.18   -37.12    -8.54 1.00     4174     2819
## Ninitial16:StageWithseedinflow:Shannon       -6.90      7.10   -20.97     7.25 1.00     4332     2922
## Ninitial32:StageWithseedinflow:Shannon        9.88      8.45    -6.82    26.82 1.00     4330     2710
## Ninitial2:StageWithoutseedinflow:Shannon    -28.24     13.36   -54.82    -2.14 1.00     3749     2924
## Ninitial4:StageWithoutseedinflow:Shannon    -22.83      7.44   -37.76    -8.22 1.00     4644     2824
## Ninitial8:StageWithoutseedinflow:Shannon    -33.92      7.55   -48.42   -19.10 1.00     3575     3007
## Ninitial16:StageWithoutseedinflow:Shannon     1.87      6.11   -10.01    14.08 1.00     4080     3018
## Ninitial32:StageWithoutseedinflow:Shannon     8.25      4.77    -0.98    17.81 1.00     4332     2988
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    17.66      0.52    16.67    18.73 1.00     7764     2080
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.3183385 0.02441864 0.2684945 0.3637714

4.2.1 Posterior predictive checks

4.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


5 Forest2

Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.

5.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               31.78      1.40    29.01    34.46 1.00     1952     2187
## StageWithoutseedinflow            28.76      1.75    25.44    32.22 1.00     2030     1803
## StageWithseedinflow:Shannon       -0.38      0.53    -1.39     0.67 1.00     1963     2089
## StageWithoutseedinflow:Shannon    -6.10      0.95    -7.99    -4.27 1.00     2030     1795
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    10.61      0.27    10.10    11.17 1.00     2507     2207
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2941946 0.02337017 0.2477921 0.3398701

5.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

5.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

5.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

5.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


5.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                80.49     10.59    60.13   101.03 1.00     3606     2797
## Ninitial4:StageWithseedinflow               101.30     12.42    77.03   125.79 1.00     3302     2784
## Ninitial8:StageWithseedinflow                70.24     17.89    35.59   104.75 1.00     3028     2896
## Ninitial16:StageWithseedinflow               84.79     29.39    27.02   141.46 1.00     3167     2752
## Ninitial32:StageWithseedinflow              116.73     43.67    32.18   202.10 1.00     3602     2997
## Ninitial2:StageWithoutseedinflow             14.01      5.89     2.36    25.49 1.00     3333     2964
## Ninitial4:StageWithoutseedinflow              3.98      5.56    -7.22    15.05 1.00     3499     2921
## Ninitial8:StageWithoutseedinflow             15.66      6.09     3.83    27.53 1.00     3955     3035
## Ninitial16:StageWithoutseedinflow             8.04      5.51    -2.60    18.93 1.00     3544     2686
## Ninitial32:StageWithoutseedinflow            13.00      7.75    -2.37    28.23 1.00     3345     2665
## Ninitial2:StageWithseedinflow:Shannon       -31.84      6.95   -45.35   -18.43 1.00     3605     2946
## Ninitial4:StageWithseedinflow:Shannon       -33.52      5.87   -45.26   -22.11 1.00     3320     2731
## Ninitial8:StageWithseedinflow:Shannon       -14.68      6.67   -27.59    -1.77 1.00     3048     2967
## Ninitial16:StageWithseedinflow:Shannon      -16.27      8.78   -33.37     0.77 1.00     3176     2774
## Ninitial32:StageWithseedinflow:Shannon      -21.81     11.11   -43.51    -0.30 1.00     3602     3009
## Ninitial2:StageWithoutseedinflow:Shannon      7.46      4.33    -1.14    15.98 1.00     3385     2859
## Ninitial4:StageWithoutseedinflow:Shannon      9.10      3.48     2.20    16.02 1.00     3482     2830
## Ninitial8:StageWithoutseedinflow:Shannon     -0.06      3.04    -6.08     5.89 1.00     3940     2992
## Ninitial16:StageWithoutseedinflow:Shannon     2.49      2.59    -2.52     7.58 1.00     3557     2681
## Ninitial32:StageWithoutseedinflow:Shannon    -0.10      3.12    -6.16     6.10 1.00     3332     2763
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     8.45      0.24     7.99     8.94 1.00     8198     2361
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4963703 0.01948423 0.4558494 0.5309853

5.2.1 Posterior predictive checks

5.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

5.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

5.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


6 Dryland

Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.

6.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow               73.89      1.28    71.32    76.35 1.00     2127     1917
## StageWithoutseedinflow            77.82      1.74    74.51    81.20 1.00     1993     2004
## StageWithseedinflow:Shannon        1.19      0.54     0.14     2.29 1.00     2179     2107
## StageWithoutseedinflow:Shannon    -1.47      1.10    -3.61     0.69 1.00     1997     1822
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.38      0.24     8.91     9.86 1.00     2810     2355
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate   Est.Error       Q2.5      Q97.5
## R2 0.01449498 0.007924524 0.00265956 0.03319514

6.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

6.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

6.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

6.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


6.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedinflow                63.34      6.75    50.23    77.03 1.00     3706     2565
## Ninitial4:StageWithseedinflow                63.18      8.77    45.82    80.06 1.00     3872     2728
## Ninitial8:StageWithseedinflow                44.69     11.12    22.80    65.94 1.00     3940     2845
## Ninitial16:StageWithseedinflow               42.05     15.88    11.27    72.96 1.00     3771     3016
## Ninitial32:StageWithseedinflow                1.33     20.14   -38.02    41.82 1.00     3492     2704
## Ninitial2:StageWithoutseedinflow             73.36      4.28    65.02    81.88 1.00     3545     2892
## Ninitial4:StageWithoutseedinflow             60.21      4.44    51.18    68.75 1.00     3758     2649
## Ninitial8:StageWithoutseedinflow             55.08      4.57    46.12    64.18 1.00     3326     2920
## Ninitial16:StageWithoutseedinflow            41.80      6.52    28.50    54.69 1.00     3867     3055
## Ninitial32:StageWithoutseedinflow            38.07      9.53    19.45    56.82 1.00     4080     3063
## Ninitial2:StageWithseedinflow:Shannon         9.15      4.65    -0.36    18.26 1.00     3734     2749
## Ninitial4:StageWithseedinflow:Shannon         7.18      4.57    -1.69    16.28 1.00     3893     2639
## Ninitial8:StageWithseedinflow:Shannon        13.34      4.59     4.64    22.45 1.00     3949     3003
## Ninitial16:StageWithseedinflow:Shannon       11.69      5.36     1.35    22.15 1.00     3763     2897
## Ninitial32:StageWithseedinflow:Shannon       21.96      5.82    10.09    33.26 1.00     3471     2647
## Ninitial2:StageWithoutseedinflow:Shannon      5.38      3.42    -1.46    12.09 1.00     3518     2823
## Ninitial4:StageWithoutseedinflow:Shannon     11.38      3.04     5.61    17.52 1.00     3861     2607
## Ninitial8:StageWithoutseedinflow:Shannon     11.95      2.74     6.51    17.44 1.00     3350     2987
## Ninitial16:StageWithoutseedinflow:Shannon    16.44      3.49     9.59    23.57 1.00     3807     2914
## Ninitial32:StageWithoutseedinflow:Shannon    16.45      4.75     6.99    25.85 1.00     4084     2953
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     7.95      0.23     7.51     8.40 1.00     7185     2888
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error     Q2.5     Q97.5
## R2 0.2235789 0.02477866 0.174982 0.2711424

6.2.1 Posterior predictive checks

6.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

6.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

6.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.